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Quadratic Functions

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alright let's talk about quadratics specifically what is a quadratic function what kind of a function is that what does quadratic mean well we already know about linear functions linear functions make a straight line and they generally have an equation that looks something like MX plus be so it's like 5x + 9 or -2/3 x + 4 4x - or something like that to make a linear function so what if we took that y equals x minus 7 and made it into y equals x squared minus 7 would it still mean Lenny Howell exponent effects dysfunction let's find out in order to make this a little bit simpler let's just drop that -7 and let's just make a graph of y equals x squared we're going to do that by making a table first and I'm going to pick some X numbers and plug them in and see what wire comes out to let's start with -3 why would we start with -3 Because I Said So if you plug -3 in 4x squared and put that -3 right here and then Square it you would get negative 3 times negative 3 write -3 times negative 3 comes out to what how it comes out to positive 9 because a negative times a negative makes a positive and if we do the same thing with negative two oh yeah by the way there's a no. -3, night if you do the same thing with the negative to the negative 2 times negative 2 with a positive for and then if we do the same thing with a negative one negative 1 times negative one is pause didn't want you can check these on a calculator if you want to go put a. At -1, one it looks kind of lennier maybe let's try 010 times 0 is still zero right so okay now it's not looking so linear anymore now after this. You'll definitely know it's not linear cuz if we plug in 1 what's 1 * 1 yep one and if I put it out at 1 100 interesting how about to what's 2 * 2 4 okay and if I do 3 * 343 squared I get 9 look what happened here it made a symmetrical little thing anywhere if we connected all the dots it's like a u shape of some kind interesting definitely not a linear function is it okay so we asked what is a quadratic function well this is a quadratic function y equals x squared and it's not the only quadratic function a quadratic function would be a n function x squared in it because any function with an x squared in it is going to make this u-shaped this curve which we call a parabola Parabola is a symmetrical U-shaped curve that happens when you graph a function with an x squared in it so what is a quadratic function a quadratic function has an x squared in it and it forms a parabola now the parabola that get from a quadratic function could be facing upward you or it could be a downward you facing thing it could actually get scooted up or scooted down or scooted to the right or scooted to the left and all of that depends on do you have y equals x squared or do you have y equals 5x squared a d i y equals negative 2/3 x squared or do you have like 1-7 or changing all these different parts of the function make it go right side up or upside down left right scooting up scooting down so how do you know what a particular function is going to look like well you have to graph it but they're all going to have a problem shape because they all an x squared and they are all quadratic okay let's say you wanted to graph y equals negative 2x squared + 7 How would how do you do it what do I do when we're just going to do what before we're going to make a table and plugs and stuff in so if I plug o my goodness I guess I'm just going to do the work for you really fast your butt in case you're wondering where I got those numbers here's where those now just came from to get this negative 11 I plugged the -3 in 4x squared and I did -3 sorry I can't write today negative 3 times negative 3 positive nine positive 9 * -2 is -18 + -18 + 7 gives you 11 that's how I got that answer and you can do that for each one of those numbers down the table what would happens when you plot these points on the graph it's going to go like this oh this one is a downward-facing parabola and it didn't start at 0-0 like the other one did okay so what if you have to do a word problem like this word problem where there's a stone being dropped from a tall bridge okay so notice on this problem it gives us an equation says the height of the stone will follow this equation it will follow this rule so rather than making you watch me do all the math I just jotted down all of the math here and if you want to check it out for yourself you can but the basic idea here again just so you are totally getting this is to figure out what goes in this blank here I figured out that when the time 0 the height would be -16 * 0 squared + 144 and then I did the math and I got 144 is my aunt which kind of makes sense because what's 6 -16 x 0 x 0 0 + 0 + 144 is 144 So eventually after doing all of that I found out that after no time the height was 144 oh that makes sense because it said the bridge is 144 ft tall and if no time has passed and it's still sitting at the top of the bridge once time is starting to pass it's going to start falling so after half a second it's at a hundred and forty feet high that means it went down a little bit which makes sense because it's falling from the bridge after 1 second it's at 128th it's falling some more after second and 1/2 its at 180% are turning to look kind of Parabola itch and after 2 seconds that it said 80 and you can see the curve starts to get steeper and steeper after 2 and 1/2 seconds that's at 44 and then after 3 seconds when I plug 03 into this equation I get -144 plus positive 144 which is 0 + that means that after 3 seconds the height of that stone is 0 feet o that means it fell all the way down to the river bed so how long did it take for the stone to reach the river 3 seconds that's when the height became 0 so you can use quadratics to solve real life problems and those real life problems happen whenever you have an x squared or in this case of t squared this gives you what we call a quadratic function so again what is a quadratic function a quadratic function is a function that has an x squared in it not just an x button x squared and when you graph it it always makes what kind of a it always forms a parabola that's all I got to say for you I will see you next time

Uploaded by Jeff Marbut.

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